The z-Transform: Introduction

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1 The z-Transform: IntroductionWhy z-Transform?Many of signals (such as x(n)=u(n), x(n) = (0.5)nu(-n), x(n) = sin(nω) etc. ) do not have a DTFT.Advantages like Fourier transform provided:Solution process reduces to a simple algebraic proceduresThe temporal domain sequence output y(n) = x(n)*h(n) can be represent as Y(z)= X(z)H(z)Properties of systems can easily be studied and characterized in z – domain (such as stability..)Topics:Definition of z –TransformProperties of z- TransformInverse z- Transform

2 Definition of the z-TransformDefinition:The z-transform of a discrete-time signal x(n) is defined bywhere z = rejw is a complex variable. The values of z for which the sum converges define a region in the z-plane referred to as the region of convergence (ROC).Notationally, if x(n) has a z-transform X(z), we writeThe z-transform may be viewed as the DTFT or an exponentially weighted sequence. Specifically, note that with z = rejw, X(z) can be looked as the DTFT of the sequence r--nx(n) and ROC is determined by the range of values of r of the following right inequation.

3 ROC & z-plane Complex z-plane Zeros and poles of X(z)z = Re(z)+jIm(z) = rejwZeros and poles of X(z)Many signals have z-transforms that are rational function of z:Factorizing it will give:The roots of the numerator polynomial, βk,are referred to as the zeros (o) and αk are referred to as poles (x). ROC of X(z) will not contain poles.

4 ROC propertiesROC is an annulus or disc in the z-plane centred at the origin. i.e.A finite-length sequence has a z-transform with a region of convergence that includes the entire z-plane except, possibly, z = 0 and z = . The point z = will be included if x(n) = 0 for n < 0, and the point z = 0 will be included if x(n) = 0 for n > 0.A right-sided sequence has a z-transform with a region of convergence that is the exterior of a circle: ROC: |z|>αA left-sided sequence has a z-transform with a region of convergence that is the interior of a circle:ROC: |z|<βThe Fourier Transform of x(n) converges absolutely if and only if ROC of z-transform includes the unit circle

12 Properties of Z-TransformLinearityIf x(n) has a z-transform X(z) with a region of convergence Rx, and if y(n) has a z-transform Y(z) with a region of convergence Ry,and the ROC of W(z) will include the intersection of Rx and Ry, that is, Rw contains .Shifting propertyIf x(n) has a z-transform X(z),Time reversalIf x(n) has a z-transform X(z) with a region of convergence Rx that is the annulus , the z-transform of the time-reversed sequence x(-n) isand has a region of convergence , which is denoted by

13 Properties of Z-TransformMultiplication by an exponentialIf a sequence x(n) is multiplied by a complex exponential αn.Convolution theormIf x(n) has a z-transform X(z) with a region of convergence Rx, and if h(n) has a z-transform H(z) with a region of convergence Rh,The ROC of Y(z) will include the intersection of Rx and Rh, that is,Ry contains Rx ∩ Rh .With x(n), y(n), and h(n) denoting the input, output, and unit-sample response, respectively, and X(z), Y(x), and H(z) their z-transforms. The z-transform of the unit-sample response is often referred to as the system function.ConjugationIf X(z) is the z-transform of x(n), the z-transform of the complex conjugate of x(n) is

14 Properties of Z-TransformDerivativeIf X(z) is the z-transform of x(n), the z-transform of isInitial value theoremIf X(z) is the z-transform of x(n) and x(n) is equal to zero for n<0, the initial value, x(0), maybe be found from X(z) as follows: